IMPROVING MATHEMATICAL REPRESENTATION SKILL BY USING PACE MODEL.
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
ME - 11
IMPROVING MATHEMATICAL REPRESENTATION SKILL
BY USING PACE MODEL
Andri Suryana
Indraprasta PGRI University
andri_16061983@yahoo.com
Abstract
Many courses in Mathematics Education Program requires a mathematical
representation skill. One of them is Mathematical Statistics. Mathematical
representation is way that used someone to express mathematical ideas visually,
mathematical expressions, or written text. However, most of the students have
difficulty in representing mathematical problems. One of models to improve
mathematical representation skill is PACE model. PACE model is a model based on
constructivist learning that has phase: Project, Activity, Cooperative Learning, and
Exercise. This model is important to apply in mathematics teaching, because it can
improve mathematical representation skill.
Keywords: Mathematical Representation Skill, PACE Model
INTRODUCTION
To learn mathematics, students need mathematical skills. One of them is Mathematical
Representation Skill. Representation Skill can help students in understanding, communicating,
and connecting mathematical concepts (Hudiono, 2005). However, mathematical representation
skill of students is very limited (NCTM, 2000), because they always use symbolic
representations. Representation can be seen as a bridge that connects part of “concrete“ and
“abstract“ in mathematics (Gravemeijer, 1994). One of courses in Mathematics Education
Program that requires Mathematical Representation Skill is Mathematical Statistics. Based on
observations at one of university in South Jakarta, the students had difficulty in changing
abstract ideas presented in problem of Mathematical Statistics to a real concept in finding
solutions.
To improve mathematical representation skill in Mathematical Statistics Course, we
need a model of learning that provides opportunities for students to be active in the classroom
and can construct the concepts to their own abilities. One model that is according to
constructivist learning theory that emphasizes on actively students is PACE model. The PACE
model was developed by Lee (1999) which is an abbreviation of Project, Activity, Cooperative
Learning and exercise. Students are taught by the PACE model is more active in the classroom
by class discussion (Lee, 1999).
Formulation of the problem in this study is how application of the model PACE in
improving Mathematical Representation Skill?. The goal of this study is to know the application
of the model PACE in improving Mathematical Representation Skill.
Because the model is very important to be applied in teaching, it will be studied in depth
in theory about the application of the PACE model in improving mathematical representation
skill. The benefit of this study, the model can be a reference in improving the quality of learning
to be more effective and efficient.
ME-79
Andri Suryana / Improving Mathematical Representation
ISBN.978-979-99314-8-1
EXPLANATION
Mathematical Representation Skill
According Cai, Lane and Jakabcsin (in Suparlan, 2005), the representation is a way to
express mathematical ideas. Variety representation is often used in communicating mathematics
are tables, drawings, graphs, mathematical expressions, and written text.
Representation relating to the interpretation. They gave different meanings according to
context in the learning process. Lesh (in Suparlan, 2005) confirms that all mathematical models
are useful, but depending on the learning objectives. Moreover, Mudzakir (2006) in his research,
classifying mathematical representation into three forms of representation, that are: 1) a visual
representation; 2) a mathematical equation or expression; and 3) The words or written text.
Based on description above, the mathematical representation skill in this study is skill in
expression or modeling of ideas, ideas or mathematical concepts in a visual form; equations or
mathematical expressions; and words or written text.
PACE Model
The PACE model was developed by Lee (1999) for statistical learning, which stands for
Project (Project), Activity (Activity), Cooperative Learning (Cooperative Learning) and exercise
(Exercise). Students are taught by the PACE model is more active in the classroom by class
discussion (Lee, 1999). PACE model is based on the principles: (1) construct their own
knowledge by guidance, (2) practice and feedback is important, and (3) active learning priority
in solving a problem. PACE model require computer technology (Lee, 1999).
In this study, the PACE model will be adapted to the characteristics of Mathematical
Statistics Course. Because the courses are rarely used computer technology, it requires more
analytic theory and more emphasis on deductive reasoning. In other words, learning PACE use
Worksheet.
The project is an important component of the PACE model. Laviatan (2008) says that
the project is an innovative form of learning that emphasizes the complex activities to solve
problems based on inquiry activities. Project activities in the form of groups. They can choose
their own topics. They were asked to find solutions to the problems of his choice. They should
make a report of the project. In this project, students are required to be active, critical and
creative. Through the project, students better understand the concepts and can improve retention
and can explore mathematical abilities, both cognitive and affective abilities.
The aim of Activity in PACE model is to introduce students to information or new
concepts by giving a task in the form of Activity Worksheet. Activity Worksheet is one form of
the Student Worksheet to learn the material. Through Activity Worksheet, students are given the
opportunity to find their own concepts that will be learned.
Cooperative learning in the PACE model is implemented in the classroom. In the study,
students work in groups and discussions to find solutions of the problem in the Discussion
Worksheet. Discussion Worksheet is a form of the Student Worksheet too to learn the material.
Through Worksheet discussion, students had the opportunity to present the findings obtained
during the discussion. During the discussion, students exchange ideas in order to get a correct
understanding of the concept.
The purpose of exercise in the PACE model is to reinforce the concepts that have been
constructed on activities and cooperative learning. This exercise is given to students in the form
of additional duties in Exercise Worksheet. Through Exercise Worksheet, students can
understand the material better. Phase exercises related to reflection, that is to re-examine the
results and the process (Polya, 1981).
ME-80
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
Based on description above, the PACE model in this study is a model of learning based
on constructivism which has a phase: Projects, Activities, and cooperative learning exercises
using Worksheet Students in the learning process.
Application of the PACE Model in Improving Mathematical Representation Skill
Application of the PACE model in improving mathematical representation skill seen from
the learning phase, that is:
1) In the activity phase, the lecturer gave Activity Worksheet for students to do at home.
At the time of the lecture, the lecturer with students discussing the Activity Worksheet.
Examples of problems in Activity Worksheet is:
To understand the function of the joint probability of two random variables,
consider the following illustration.
Two balls are drawn from a bag containing 3 black balls, 2 yellow balls and 3
red balls. Let X and Y is the number of black balls and yellow balls are drawn.
a) Determine the possible values of the random variables X and Y.
b) Determine the joint probability function, p x, y .
c)
Make a chart of the joint probability distribution, p x, y .
d)
Compute
p x, y .
x
y
2) In cooperative learning phase, the lecturer gave Discussion Worksheet to each group.
This is a continuation of Activity Worksheet that has a higher level of difficulty.
Examples of problems in Discussion Worksheet is:
Suppose the function h(x,y) has the form:
1
; 0 y x 1
h x, y x
0; x, y other
Describe the domain of the function, then if X and Y are stochastically
independent random variables? Describe your answer.
3) In the exercise phase, the lecturer gives an additional task in Exercise Worksheet to
reinforce the concepts that have been constructed in activity and cooperative learning.
Through this phase, students are asked to try various types of questions that reinforce
the concepts. Examples of problems in Exercise Worksheet is:
ME-81
Andri Suryana / Improving Mathematical Representation
ISBN.978-979-99314-8-1
Suppose joint probability distribution s x, y given in the following table.
X
Determine the value of
0
1
2
3
0
0,05
0,10
0,10
0,05
2
3
y 0
x 0
Y
1
0,05
0,25
0,15
0,05
2
0,10
0,05
0,05
0
s x, y and s x, y .
4) In the project phase, the lecturer gives Project Worksheet. Project activities in the form
of groups. They can choose their own topics. They were asked to find solutions to the
problems of his choice. They should make a report of the project. Examples of
problems in Project Worksheet is:
PROJECT TASK
Topic Problem: Markov Chain
1.
2.
3.
4.
5.
Instructions:
Each group should choose one Markov Chain applications, for example the
Markov chain in displacement brands, shopping, a game of snakes and
ladders, weather forecasts, stock price fluctuations, and others.
Each group conducted an investigation directly into the field by Markov
Chain applications selected.
Understand the questions in this project assignment. If you experience
difficulty, please consult with the lecturer.
Make a report on a group project assignments and collected in the last
lecture.
Each group should present the results of the project tasks.
The questions:
1. Describe the Background of Markov Chain Application.
2. Describe the Markov Chain.
3. Describe the Probability Matrix Transition. Give an examples.
4. Describe the n-step transition probability.
5. Describe the Chapman Kolmogorov equation along with its proof. Give an
example.
6. Describe the state vector and steady state conditions. Give an examples.
7. Describe the Markov Chain Applications which your group choose.
8. Describe the research methods used in this project.
9. Construct a table of raw data directly from the results of the investigation.
10. Construct a table of transition probability.
11. Develop a query based on the tables that were made previously regarding
ME-82
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
the conditional probability, conditional expectation, and conditional
variance.
12. Describe the discussion based on the results of the investigation directly
from the field and the tables were made previously.
13. Provide conclusions and suggestions based on the discussion above.
Write the libraries that you use in the task group 's project.
Based on the above steps, shown theoretically that the PACE model can be applied to
improve the mathematical representation skill in Mathematical Statistics Course.
CONCLUSION AND SUGGESTION
Students are required to explore the potential of mathematical representations in
learning Mathematics Statistics Courses. One of the models in order to improve the ability of
the PACE model is a mathematical representation. PACE model is one based on constructivist
learning model that has phase: Project, Activity, Cooperative Learning), and exercise. Through
this study, expected to be developed in the direction of further research.
BIBLIOGRAPHY
Gravemeijer, K.P.E. (1994). Developing Realistic Mathematics Education . Utrecht: Feudenthal
Institution.
Herrhyanto, Nar & Tuti G. (2009 ). Introduction to Mathematical Statistics. New York: Widya
Yrama.
Hudiono, B. (2005). Multi Role of Learning Discourse Representation the Development of
Mathematical Ability and Power Representation in junior high students . Dissertation.
PPS UPI Bandung: Not published.
Laviatan, T. (2008). Innovative Teaching and Assessment Method: QBI and Project Based
Learning. Mathematics Education Research Journal, Vol 10 , 2 , 105-116.
Lee, Carl. (1999). An Assessment of the PACE Strategy for an introduction statistics Course.
USA: Central Michigan University.
Mudzakir, A. (2006). Educational Psychology. New York: Library loyal.
National Council of Teachers of Mathematics (2000). Principles and Standards for School
Mathematics. [ Online ].
Available: http://krellinst.org/AiS/textbook/Manual/stand/NCTM_ stand.html. [20 June
2012] .
Polya, G. (1981). Mathematical Discovery: On Understanding, Learning, and Teaching Problem
Solving. New York: John Wiley Inc.
Ross, S. M. (2000). Introduction to Probability Models. Ed. 7th. San Diego: Academic Press .
ME-83
Andri Suryana / Improving Mathematical Representation
ISBN.978-979-99314-8-1
Suparlan, A. (2005). Problem Based Learning for Developing Comprehension Ability and
Mathematical Representation of Junior High School Students. Thesis. UPI :
Unpublished.
ME-84
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
ME - 11
IMPROVING MATHEMATICAL REPRESENTATION SKILL
BY USING PACE MODEL
Andri Suryana
Indraprasta PGRI University
andri_16061983@yahoo.com
Abstract
Many courses in Mathematics Education Program requires a mathematical
representation skill. One of them is Mathematical Statistics. Mathematical
representation is way that used someone to express mathematical ideas visually,
mathematical expressions, or written text. However, most of the students have
difficulty in representing mathematical problems. One of models to improve
mathematical representation skill is PACE model. PACE model is a model based on
constructivist learning that has phase: Project, Activity, Cooperative Learning, and
Exercise. This model is important to apply in mathematics teaching, because it can
improve mathematical representation skill.
Keywords: Mathematical Representation Skill, PACE Model
INTRODUCTION
To learn mathematics, students need mathematical skills. One of them is Mathematical
Representation Skill. Representation Skill can help students in understanding, communicating,
and connecting mathematical concepts (Hudiono, 2005). However, mathematical representation
skill of students is very limited (NCTM, 2000), because they always use symbolic
representations. Representation can be seen as a bridge that connects part of “concrete“ and
“abstract“ in mathematics (Gravemeijer, 1994). One of courses in Mathematics Education
Program that requires Mathematical Representation Skill is Mathematical Statistics. Based on
observations at one of university in South Jakarta, the students had difficulty in changing
abstract ideas presented in problem of Mathematical Statistics to a real concept in finding
solutions.
To improve mathematical representation skill in Mathematical Statistics Course, we
need a model of learning that provides opportunities for students to be active in the classroom
and can construct the concepts to their own abilities. One model that is according to
constructivist learning theory that emphasizes on actively students is PACE model. The PACE
model was developed by Lee (1999) which is an abbreviation of Project, Activity, Cooperative
Learning and exercise. Students are taught by the PACE model is more active in the classroom
by class discussion (Lee, 1999).
Formulation of the problem in this study is how application of the model PACE in
improving Mathematical Representation Skill?. The goal of this study is to know the application
of the model PACE in improving Mathematical Representation Skill.
Because the model is very important to be applied in teaching, it will be studied in depth
in theory about the application of the PACE model in improving mathematical representation
skill. The benefit of this study, the model can be a reference in improving the quality of learning
to be more effective and efficient.
ME-79
Andri Suryana / Improving Mathematical Representation
ISBN.978-979-99314-8-1
EXPLANATION
Mathematical Representation Skill
According Cai, Lane and Jakabcsin (in Suparlan, 2005), the representation is a way to
express mathematical ideas. Variety representation is often used in communicating mathematics
are tables, drawings, graphs, mathematical expressions, and written text.
Representation relating to the interpretation. They gave different meanings according to
context in the learning process. Lesh (in Suparlan, 2005) confirms that all mathematical models
are useful, but depending on the learning objectives. Moreover, Mudzakir (2006) in his research,
classifying mathematical representation into three forms of representation, that are: 1) a visual
representation; 2) a mathematical equation or expression; and 3) The words or written text.
Based on description above, the mathematical representation skill in this study is skill in
expression or modeling of ideas, ideas or mathematical concepts in a visual form; equations or
mathematical expressions; and words or written text.
PACE Model
The PACE model was developed by Lee (1999) for statistical learning, which stands for
Project (Project), Activity (Activity), Cooperative Learning (Cooperative Learning) and exercise
(Exercise). Students are taught by the PACE model is more active in the classroom by class
discussion (Lee, 1999). PACE model is based on the principles: (1) construct their own
knowledge by guidance, (2) practice and feedback is important, and (3) active learning priority
in solving a problem. PACE model require computer technology (Lee, 1999).
In this study, the PACE model will be adapted to the characteristics of Mathematical
Statistics Course. Because the courses are rarely used computer technology, it requires more
analytic theory and more emphasis on deductive reasoning. In other words, learning PACE use
Worksheet.
The project is an important component of the PACE model. Laviatan (2008) says that
the project is an innovative form of learning that emphasizes the complex activities to solve
problems based on inquiry activities. Project activities in the form of groups. They can choose
their own topics. They were asked to find solutions to the problems of his choice. They should
make a report of the project. In this project, students are required to be active, critical and
creative. Through the project, students better understand the concepts and can improve retention
and can explore mathematical abilities, both cognitive and affective abilities.
The aim of Activity in PACE model is to introduce students to information or new
concepts by giving a task in the form of Activity Worksheet. Activity Worksheet is one form of
the Student Worksheet to learn the material. Through Activity Worksheet, students are given the
opportunity to find their own concepts that will be learned.
Cooperative learning in the PACE model is implemented in the classroom. In the study,
students work in groups and discussions to find solutions of the problem in the Discussion
Worksheet. Discussion Worksheet is a form of the Student Worksheet too to learn the material.
Through Worksheet discussion, students had the opportunity to present the findings obtained
during the discussion. During the discussion, students exchange ideas in order to get a correct
understanding of the concept.
The purpose of exercise in the PACE model is to reinforce the concepts that have been
constructed on activities and cooperative learning. This exercise is given to students in the form
of additional duties in Exercise Worksheet. Through Exercise Worksheet, students can
understand the material better. Phase exercises related to reflection, that is to re-examine the
results and the process (Polya, 1981).
ME-80
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
Based on description above, the PACE model in this study is a model of learning based
on constructivism which has a phase: Projects, Activities, and cooperative learning exercises
using Worksheet Students in the learning process.
Application of the PACE Model in Improving Mathematical Representation Skill
Application of the PACE model in improving mathematical representation skill seen from
the learning phase, that is:
1) In the activity phase, the lecturer gave Activity Worksheet for students to do at home.
At the time of the lecture, the lecturer with students discussing the Activity Worksheet.
Examples of problems in Activity Worksheet is:
To understand the function of the joint probability of two random variables,
consider the following illustration.
Two balls are drawn from a bag containing 3 black balls, 2 yellow balls and 3
red balls. Let X and Y is the number of black balls and yellow balls are drawn.
a) Determine the possible values of the random variables X and Y.
b) Determine the joint probability function, p x, y .
c)
Make a chart of the joint probability distribution, p x, y .
d)
Compute
p x, y .
x
y
2) In cooperative learning phase, the lecturer gave Discussion Worksheet to each group.
This is a continuation of Activity Worksheet that has a higher level of difficulty.
Examples of problems in Discussion Worksheet is:
Suppose the function h(x,y) has the form:
1
; 0 y x 1
h x, y x
0; x, y other
Describe the domain of the function, then if X and Y are stochastically
independent random variables? Describe your answer.
3) In the exercise phase, the lecturer gives an additional task in Exercise Worksheet to
reinforce the concepts that have been constructed in activity and cooperative learning.
Through this phase, students are asked to try various types of questions that reinforce
the concepts. Examples of problems in Exercise Worksheet is:
ME-81
Andri Suryana / Improving Mathematical Representation
ISBN.978-979-99314-8-1
Suppose joint probability distribution s x, y given in the following table.
X
Determine the value of
0
1
2
3
0
0,05
0,10
0,10
0,05
2
3
y 0
x 0
Y
1
0,05
0,25
0,15
0,05
2
0,10
0,05
0,05
0
s x, y and s x, y .
4) In the project phase, the lecturer gives Project Worksheet. Project activities in the form
of groups. They can choose their own topics. They were asked to find solutions to the
problems of his choice. They should make a report of the project. Examples of
problems in Project Worksheet is:
PROJECT TASK
Topic Problem: Markov Chain
1.
2.
3.
4.
5.
Instructions:
Each group should choose one Markov Chain applications, for example the
Markov chain in displacement brands, shopping, a game of snakes and
ladders, weather forecasts, stock price fluctuations, and others.
Each group conducted an investigation directly into the field by Markov
Chain applications selected.
Understand the questions in this project assignment. If you experience
difficulty, please consult with the lecturer.
Make a report on a group project assignments and collected in the last
lecture.
Each group should present the results of the project tasks.
The questions:
1. Describe the Background of Markov Chain Application.
2. Describe the Markov Chain.
3. Describe the Probability Matrix Transition. Give an examples.
4. Describe the n-step transition probability.
5. Describe the Chapman Kolmogorov equation along with its proof. Give an
example.
6. Describe the state vector and steady state conditions. Give an examples.
7. Describe the Markov Chain Applications which your group choose.
8. Describe the research methods used in this project.
9. Construct a table of raw data directly from the results of the investigation.
10. Construct a table of transition probability.
11. Develop a query based on the tables that were made previously regarding
ME-82
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
the conditional probability, conditional expectation, and conditional
variance.
12. Describe the discussion based on the results of the investigation directly
from the field and the tables were made previously.
13. Provide conclusions and suggestions based on the discussion above.
Write the libraries that you use in the task group 's project.
Based on the above steps, shown theoretically that the PACE model can be applied to
improve the mathematical representation skill in Mathematical Statistics Course.
CONCLUSION AND SUGGESTION
Students are required to explore the potential of mathematical representations in
learning Mathematics Statistics Courses. One of the models in order to improve the ability of
the PACE model is a mathematical representation. PACE model is one based on constructivist
learning model that has phase: Project, Activity, Cooperative Learning), and exercise. Through
this study, expected to be developed in the direction of further research.
BIBLIOGRAPHY
Gravemeijer, K.P.E. (1994). Developing Realistic Mathematics Education . Utrecht: Feudenthal
Institution.
Herrhyanto, Nar & Tuti G. (2009 ). Introduction to Mathematical Statistics. New York: Widya
Yrama.
Hudiono, B. (2005). Multi Role of Learning Discourse Representation the Development of
Mathematical Ability and Power Representation in junior high students . Dissertation.
PPS UPI Bandung: Not published.
Laviatan, T. (2008). Innovative Teaching and Assessment Method: QBI and Project Based
Learning. Mathematics Education Research Journal, Vol 10 , 2 , 105-116.
Lee, Carl. (1999). An Assessment of the PACE Strategy for an introduction statistics Course.
USA: Central Michigan University.
Mudzakir, A. (2006). Educational Psychology. New York: Library loyal.
National Council of Teachers of Mathematics (2000). Principles and Standards for School
Mathematics. [ Online ].
Available: http://krellinst.org/AiS/textbook/Manual/stand/NCTM_ stand.html. [20 June
2012] .
Polya, G. (1981). Mathematical Discovery: On Understanding, Learning, and Teaching Problem
Solving. New York: John Wiley Inc.
Ross, S. M. (2000). Introduction to Probability Models. Ed. 7th. San Diego: Academic Press .
ME-83
Andri Suryana / Improving Mathematical Representation
ISBN.978-979-99314-8-1
Suparlan, A. (2005). Problem Based Learning for Developing Comprehension Ability and
Mathematical Representation of Junior High School Students. Thesis. UPI :
Unpublished.
ME-84